Cross-correlation of signals using event-based sampling

ABSTRACT

A sub-circuit for facilitating the synchronization of event-based samples of signals in a cross-correlation circuit utilizing event-based sampling is provided. The sub-circuit alternatively integrates one of the signals to be cross-correlated and alternates between the signals in response to the output of a hysteretic comparator. The invention extends to a method of manipulating the input signals to a cross-correlation circuit utilizing event-based sampling so-as to facilitate the synchronization of the event-based samples of the signals.

The U.S. Government has a paid-up license in this invention and theright in limited circumstances to require the patent owner to licenseothers on reasonable terms as provided for by the terms of Grant No. EEC9731478 and Award No. 0440217, awarded by the National ScienceFoundation (NSF), and by the terms of Contract No. 9080 awarded by theDefense Advanced Research Projects Agency (DARPA).

FIELD OF THE INVENTION

This invention relates to the cross-correlation of signals. Inparticular, but not exclusively, the invention relates to thesynchronization of signals in cross-correlation circuits employingevent-based or integrative sampling. The invention further relates to amethod of synchronizing event-based samples of signals in across-correlation system employing event-based or integrative sampling.

BACKGROUND TO THE INVENTION

Cross correlation is a mathematical operation in which two signals (inthe time domain) are compared with one another, usually over a fixedlength of each signal. The comparison returns a third signal, which isnormally the magnitude of the similarity between the two signals, atvarious delays between the two signals. This is expressed mathematicallyas:

$\begin{matrix}{{R_{fg}(\tau)} = {\int_{{- T}/2}^{T/2}{{f(t)}{g( {t + \tau} )}{\mathbb{d}t}}}} & (1)\end{matrix}$where R_(fg)(τ) is the cross-correlation function at delay τ for the twosignals f(t) and g(t) evaluated over the time interval −T/2 to T/2.

Cross-correlation is extremely important in modern telecommunicationssystems, mainly due it being a key operation in a class of modulationsystems called Code-Division Multiple Access (CDMA) systems. CDMAsystems make very efficient use of the available telecommunicationsspectra, and are therefore at the time of this application the firstchoice in most wireless data systems, including GSM (Global System forMobile communication), HSDPA (High-speed Download Packet Access) andvarious other widely used mobile data networks. The principles of CDMAare also used, in somewhat different form, in GNSS (Global NavigationSatellite Systems) networks such as the GPS (Global Positioning System)network and the GLONASS (Global Navigation Satellite System) network.

The signals transmitted over a CDMA system are encoded beforetransmission, in such a way as to spread their energy over a widerspectral bandwidth than that used by the raw signal. This improves theirnoise immunity and allows several signals to share the same spectralband. At the core of all CDMA receivers is a processing system that mustcross-correlate the encoded incoming signals with local copies of thecodes which were used to encode the transmission. This cross-correlationis effectively the decoding operation, Cross-correlating an encodedsignal with the original code used to encode it reveals the original rawsignal. Cross-correlation is therefore an essential operation in allCDMA systems.

Conventional cross-correlation is, however, a computationally intensiveoperation, particularly in mobile systems, and according consumes largeamounts of energy, a valuable commodity in most mobile devices. Movingtransmitters or receivers send or receive Doppler-shifted signals (theseare signals whose frequencies are altered by the relative motion betweentransmitter and receiver). The demodulation (decoding) of the signalwill not work if only one of the two signals to be correlated has asignificant Doppler shift. It therefore becomes necessary to run severalcross-correlations in parallel, using a variety of artificiallyDoppler-shifted versions of the local code. This is particularly aproblem in GPS systems. A typical GPS system may need to decode withrespect to 32 codes, and with up to 20 Doppler shifted variants of eachcode, meaning that over 600 correlation operations, must run inparallel. This is a core reason why integration of GPS systems intomobile phones, for example, has been slow: the GPS receivers consume alarge amount of power.

It has been estimated that up to 40% of the power usage in a handheldGPS device can be attributed to cross-correlation operations.

The standard approach to cross-correlation is digital signal processing(DSP). The incoming signal is sampled (digitized) and stored in memory.The cross-correlation is then calculated mathematically and numerically.Such calculations are typically slow, intrinsically serial in nature,and consume a large amount of power.

Neuromorphic systems are engineering systems that are based onbiological neural principles. They endeavor to copy the successfulmethods which have evolved in living organisms. Living organisms make agreat deal of use of cross-correlation. For example, to estimate themotion of an object which is seen in the visual field, or to estimatethe position from where sound originates, requires cross-correlation ofvisual or auditory signals.

Insect eyes, which consist of multiple individual segments, have alreadybeen copied to develop a useful electronic cross-correlator. They havean intrinsic pixel-by-pixel structure which translates well into digitalsystems. The technology is called a RAKE array and forms the basis ofsome commercially available GPS receivers.

Mammalian vision and auditory systems do not use the same approach asinsects. It is not yet known how they cross-correlate, but structuresequivalent to the insect systems have been searched for and, to the bestof the applicant's knowledge, have not been found.

A model for mammalian cross-correlation which makes use of neuralstructures called half-center oscillators, which are widespread in themammalian nervous system, has been proposed.

One advantage of this approach is that it is extremely robust to noise.Mammalian neural systems make use of very low electrical potentiallevels, and have chemical transmission paths that are inherentlystochastic. This means that any system used in mammals will beintrinsically robust to noise. This advantage is one of the keys to theneuromorphic engineering approach. There has been a consistent trendtowards the use of lower voltages in integrated circuits, as lowervoltage circuits require less power. However, as the voltages get lower,the role of noise in circuits becomes more significant. The neuromorphicapproach is that, instead of trying to mitigate noise, one should startwith a process that is inherently robust to noise, and that suchprocesses can be found in biological systems.

The basis of the present invention lies in the method of sampling thesignal. In conventional signal processing, a time series is created froman analog signal by quantizing (measuring) the signal at evenlydistributed times; the separation of the times is referred to as thesampling rate. If the signal is sampled at a sufficiently high samplingrate, and with sufficient accuracy in amplitude, then the resulting timeseries is considered to be an accurate representation of the originalsignal. This is the basic operation with which most digital signalprocessing starts.

An alternative way of sampling a signal is referred to as event-based orintegral sampling. In event-based sampling, the analog signal is fedinto an integrator circuit, which integrates the signal as well asassociated noise. The slope of the output of the integrator can bealtered by also adding a constant, or drift, term. Mathematically theintegrator output is defined as follows:

$\begin{matrix}{{v(\tau)} = {{\int_{t_{0}}^{t_{0} + \tau}m} + \sigma + {{g(t)}\ {\mathbb{d}t}}}} & (2)\end{matrix}$where τ is the interval since the start of integration, v(τ) is theintegrator output voltage, m is the drift term, σ is a noise parameter(for example, the standard deviation of some additive Gaussian whitenoise), and g(t) is the signal.

When this integrated signal reaches some predefined threshold voltage θ,the integrator is reset to zero and a new integration interval begins.This event is often referred to as “firing” or “spiking”.

The information generated by this process is a sequence of firing timest₀, t₁, . . . t_(n). It has been shown that if the system isappropriately set up, this sequence of times represents the analog inputsignal without a loss of information. Whereas the conventional samplingprocess converts the analog signal into a series of varying amplitudesat fixed times (a time series), event-based sampling (also referred toas integrate-and-fire sampling) converts the signal into a series offixed events at varying times (this is referred to as an event series).

The mathematical cross-correlation expressed in Eq. 1 above isstraightforward to implement if the two signals to be cross-correlatedare available as two time series. To cross-correlate the information inevent series form is more difficult because, while two times serieswhich are sampled on the same time basis are intrinsically synchronized,the same cannot be said for two event series. The basis of event-basedsampling is that the time of the event encodes the signal, so it is notpossible to synchronize the times without somehow altering theinformation.

OBJECT OF THE INVENTION

It is an object of this invention to provide a sub-circuit and methoduseful for synchronizing event-based samples of signals to becross-correlated in a cross-correlation circuit utilizing event-basedsampling, which will at least partially alleviate some of theabovementioned problems.

SUMMARY OF THE INVENTION

In accordance with this invention there is provided a synchronizationsub-circuit for a cross-correlation circuit utilizing event-basedsampling, comprising

-   -   a switch having first and second input terminals for receiving        first and second input signals to be cross-correlated, a control        terminal and an output terminal;    -   an integrator having an input terminal connected to the output        terminal of the switch, and an output terminal providing an        output signal; and    -   a hysteretic comparator having an input terminal connected to        the output terminal of the integrator, and an output terminal        providing an output signal that switches between a first state        and a second state in response to the output signal of the        integrator, and wherein the output terminal of the hysteretic        comparator is connected to the control terminal of the switch        causing the switch to alternately relay the first and the second        signals to the integrator input terminal with an alternation        occurring in response to each transition of the output signal of        the hysteretic comparator between the first and the second        state.

Further features of the invention provide for the integrator to be anoperational amplifier having a capacitor connected between a negativeinput terminal and an output terminal thereof; for the hystereticcomparator to be a Schmidt trigger; and for the hysteretic comparator todefine a higher and a lower threshold value.

Still further features of the invention provide for the first and secondinput signals to include drift terms causing the integrator to integratein opposite directions, depending on which of the first and second inputsignals are being integrated; for the drift term associated with thefirst input signal to cause the integrator to integrate upwards in thedirection of positive voltage when the first input signal is beingintegrated and for the drift term associated with the second inputsignal to cause the integrator to integrate downwards in the directionof negative voltage when the second input signal is being integrated;for the switch to relay the first input signal to the integrator whenthe output signal of the comparator is in the first state and the secondinput signal when the output of the comparator is in the second state;and for the output signal of the comparator to change from the firststate to the second state when the integrator output signal rises abovethe higher threshold and from the second state to the first state whenthe integrator output drops below the lower threshold.

In accordance with a further aspect of this invention there is provideda method of manipulating two input signals to a cross-correlationcircuit utilizing event-based sampling, comprising the steps of

-   -   relaying a first of the two input signals to an integrator when        a control signal is in a first state;    -   integrating the first input signal in a direction of positive        voltage when the control signal is in the first state;    -   relaying a second of the two input signals to the integrator        when the control signal is in a second state;    -   integrating the second input signal in a direction of negative        voltage when the control signal is in the second state;    -   changing the control signal from the first state to the second        state when an output of the integrator rises above a higher        threshold value; and    -   changing the control signal from the second state to the first        state when the output of the integrator drops below a lower        threshold value.

Further features of the invention provide for the method to include thesteps of adding a first drift term to the first input signal causing theintegrator to integrate in the direction of positive voltage when thefirst input signal is being integrated; adding a second drift term tothe second input signal causing the integrator to integrate in thedirection of negative voltage when the second input voltage is beingintegrated; and connecting the output of the integrator to a Schmidttrigger defining the higher and lower threshold values and the output ofwhich provides the control signal.

In accordance with a still further aspect of this invention there isprovided a method of manipulating two input signals to across-correlation circuit utilizing event-based sampling, comprising thesteps of

-   -   integrating a first of the two input signals in a direction of        positive voltage while a control signal remains in a first        state;    -   integrating a second of the two input signals in a direction of        negative voltage while the control signal remains in a second        state;    -   switching the control signal from the first state to the second        state when a result of the integration rises above a higher        predetermined threshold while the control signal is in the first        state; and    -   switching the control signal from the second to the first state        when the result of the integration drops below a lower        predetermined threshold while the control signal is in the        second state.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will now be described, by way of example only withreference to the accompanying representations in which:

FIG. 1 is a schematic circuit diagram of a synchronization sub-circuitin accordance with the invention; and

FIG. 2 is a timing diagram exemplifying the signals obtained at variouspositions in the sub-circuit of FIG. 1.

DETAILED DESCRIPTION WITH REFERENCE TO THE DRAWINGS

A sub-circuit (1) in accordance with the invention is shown in FIG. 1and includes an integrator (3) which is set up to integrate twodifferent combination input signals (5 and 7) by sequentially switchingbetween them.

The integrator (3) is formed in a conventional way by means of anoperational amplifier (9) having a capacitor (11) connected between itsnegative input terminal (13) and output terminal (15). The input signals(5 and 7) to the integrator (3) are fed in through a switch (17) whichselects either one of the two combination input signals (5 and 7) to berelayed to the integrator (3). The selection is controlled by the outputof a hysteretic comparator (19), in this example an inverting Schmitttrigger. The first of the input signals (5) consists of f(t) (a signalto be cross correlated with g(t) as in Eq. 1 above) combined with adrift term m and some noise σ. The second input signal (7) consists of−g(t) (the negative of g(t)) combined with a different drift term n andnoise σ. The noise may be different or identical for both combinationinput signals (5 and 7).

For the purpose of this description it is assumed that f(t) and g(t)both have a zero mean value, as is usual in this type of signalprocessing; although that is by no means necessary for the circuit tofunction correctly. The drift terms m and n are chosen and implementedto cause the integrator (3) to integrate in opposite directions, so thatfor m the integrator (3) integrates upward (in the direction of positivevoltage polarity) and for n the integrator (3) integrates downward (inthe direction of negative polarity).

The output v(t) of the integrator (3) is fed into the input terminal(21) of the inverting Schmitt trigger P (19). A Schmitt trigger, orhysteretic comparator, is a well-known circuit element. It defines twothreshold levels separated by a hysteresis gap or band. At any moment,one of the threshold levels, either higher or lower, is in operation. Ifthe higher threshold is in operation, the output of the Schmitt triggerwill change state when the input exceeds the higher threshold level. Atthat time, the comparator will switch to using the other, lower,threshold level. The output will not change state again until the inputtraverses below the lower threshold, at which time the threshold inoperation will change back to the higher level.

The invention is further described by means of example signals in FIG.2. The signals are vertically separated for clarity and shows the firstinput signal f(t) to be cross-correlated, the second input signal to becross-correlated g(t), the noise signal σ, the integrator output signalv(t), and the Schmitt trigger output signal q(t). The two thresholds forthe Schmitt trigger are indicated by the horizontal dotted lines (23 and25). It can be seen that the integrator integrates upwards in thedirection of positive voltage, integrating the first combination inputsignal (5) (f(t)+m +σ), until the higher threshold (23) of the Schmidttrigger (19) is reached at approximately 200 ms. At that point, theSchmidt trigger (19) changes state, as is evident from its output q(t),and the input is switched to integrate the second combination inputsignal (7) (−g(t)+n+σ). The second signal (7) is then integrateddownwards (in the direction of negative voltage) until the lowerthreshold (25) of the Schmidt trigger (19) is transitioned at about time520 ms. At that point the switch (17) again switches to the first inputsignal (5) and the process then repeats itself, with further transitionevents as indicated by the arrows (27).

The working of the sub-circuit (1) depends on the fact that the startpoint of the integration of one signal is triggered by the end point ofthe integration of the other. In other words, the precise event in onesignal stream triggers the start of sampling of the next event in theother. This has the effect of synchronizing the event-based sampling ofthe two signals, in a way that is appropriate for furthercross-correlating.

It should be noted that in the absence of noise, it is foreseeable thatthe integrations would become phase-locked to a strong input signal suchas those shown in FIG. 2. The noise serves to scatter the samplingthrough a larger range of times It has been found that the sub-circuit'sperformance improves up to a certain level with added noise, after whichit deteriorates. In this respect its behavior is similar to the physicalphenomenon known as stochastic resonance. In certain cases it is evenforeseeable that a certain level of noise may be added to the inputsignals. The noise may be added to the signals as indicated in FIG. 2or, alternatively, at any suitable location in the circuit.

At this stage it should be noted that the input series used for thecross-correlation of the signals are the series of intervals betweenevents (27), as indicated by the labels τf_(i) and τg_(i) in FIG. 2,where i is an integer number (0, 1, 2, . . . ). It should be appreciatedthat these intervals form two separate series that may be processedseparately. The process is simple; the intervals are simply histogrammedor binned, i.e. a histogram is developed in which the height of a columnfor interval x is the number of intervals for which the length was x.The histogram for the series τf_(i) above will have the shaperepresented by the following equation:

$\begin{matrix}{{H_{\tau\; f_{i}}(\tau)} = {{K\lbrack {\frac{\theta}{\sqrt{2{\pi\sigma}^{2}\tau^{3}}}{\exp( {- \frac{( {\theta - {m\;\tau}} )^{2}}{2\;\sigma^{2}\tau}} )}} \rbrack}{R_{fg}(\tau)}}} & (3)\end{matrix}$where K is a scaling factor which depends on the number of intervalsaccumulated, θ is the potential difference between thresholds, and allother symbols have their meaning as before. The desiredcross-correlation function R_(fg)(τ) may be established by simplydividing the histogram by the expression in square brackets in equation(3) above. Similarly, the histogram for the series τg_(i) may have theshape represented by the following equation:

$\begin{matrix}{{H_{\tau\; g_{i}}(\tau)} = {{K\lbrack {\frac{\theta}{\sqrt{2\;\pi\;\sigma^{2}\tau^{3}}}{\exp( {- \frac{( {\theta - {n\;\tau}} )^{2}}{2\sigma^{2}\tau}} )}} \rbrack}{R_{gf}(\tau)}}} & (4)\end{matrix}$

The sub-circuit (1) of the invention described above may therefore beused as a fundamental building block in cross-correlation circuits whichmake use of event-based or integrative sampling. It should be noted thatthe sub-circuit (1) intrinsically synchronizes the event-based samplesfor the two input signals; it produces as an output a set of intervalswhich can easily be post-processed into histograms which show thecross-correlation function which is required; and its operation is notsignificantly affected by the presence of noise and in fact is improvedby moderate levels of noise. It is therefore more suitable thanconventional circuits for low-noise electronics.

Numerous modifications may be made to the embodiment of the inventiondescribed above without departing from the scope of the invention. Inparticular, the circuit elements such as the switch, integrator andcomparator may be replaced with alternative components providing similarfunctionality.

1. A synchronization sub-circuit for a cross-correlation circuitutilizing event-based sampling, comprising a switch having first andsecond input terminals for receiving first and second input signals tobe cross-correlated, a control terminal and an output terminal; anintegrator having an input terminal connected to the output terminal ofthe switch, and an output terminal providing an output signal; and ahysteretic comparator having an input terminal connected to the outputterminal of the integrator, and an output terminal providing an outputsignal that switches between a first state and a second state inresponse to the output signal of the integrator, and wherein the outputterminal of the hysteretic comparator is connected to the controlterminal of the switch causing the switch to alternately relay the firstand the second signals to the integrator input terminal with analternation occurring in response to each transition of the outputsignal of the hysteretic comparator between the first and the secondstate.
 2. A synchronization sub-circuit as claimed in claim 1 in whichthe integrator is an operational amplifier having a capacitor connectedbetween a negative input terminal and an output terminal thereof.
 3. Asynchronization sub-circuit as claimed in claim 1 in which thehysteretic comparator is a Schmidt trigger.
 4. A synchronizationsub-circuit as claimed in claim 1 in which the first and second inputsignals include drift terms causing the integrator to integrate inopposite directions, depending on which of the first and second inputsignals are being integrated.
 5. A synchronization sub-circuit asclaimed in claim 4 in which the drift term associated with the firstinput signal causes the integrator to integrate upwards in the directionof positive voltage when the first input signal is being integrated andfor the drift term associated with the second input signal to cause theintegrator to integrate downwards in the direction of negative voltagewhen the second input signal is being integrated.
 6. A synchronizationsub-circuit as claimed in claim 1 in which the switch relays the firstinput signal to the integrator when the output signal of the comparatoris in the first state and the second input signal when the output of thecomparator is in the second state.
 7. A synchronization sub-circuit asclaimed in claim 6 in which the hysteretic comparator defines a higherand a lower threshold value.
 8. A synchronization sub-circuit as claimedin claim 7 in which the output signal of the comparator changes from thefirst state to the second state when the integrator output signal risesabove the higher threshold and from the second state to the first statewhen the integrator output drops below the lower threshold.
 9. A methodof manipulating two input signals to a cross-correlation circuitutilizing event-based sampling, comprising the steps of relaying a firstof the two input signals to an integrator when a control signal is in afirst state; integrating the first input signal in a direction ofpositive voltage when the control signal is in the first state; relayinga second of the two input signals to the integrator when the controlsignal is in a second state; integrating the second input signal in adirection of negative voltage when the control signal is in the secondstate; changing the control signal from the first state to the secondstate when an output of the integrator rises above a higher thresholdvalue; and changing the control signal from the second state to thefirst state when the output of the integrator drops below a lowerthreshold value.
 10. A method as claimed in claim 9 including the stepsof adding a first drift term to the first input signal causing theintegrator to integrate in the direction of positive voltage when thefirst input signal is being integrated and adding a second drift term tothe second input signal causing the integrator to integrate in thedirection of negative voltage when the second input signal is beingintegrated.
 11. A method as claimed in claim 10 including the step ofconnecting the output of the integrator to a Schmidt trigger definingthe higher and lower threshold values and the output of which providesthe control signal.
 12. A method of manipulating two input signals to across-correlation circuit utilizing event-based sampling, comprising thesteps of integrating a first of the two input signals in a direction ofpositive voltage while a control signal remains in a first state;integrating a second of the two input signals in a direction of negativevoltage while the control signal remains in a second state; switchingthe control signal from the first state to the second state when aresult of the integration rises above a higher predetermined thresholdwhile the control signal is in the first state; and switching thecontrol signal from the second to the first state when the result of theintegration drops below a lower predetermined threshold while thecontrol signal is in the second state.